Skok in his Etimologijski rječnik devotes a whole page to the entry ćutjeti, from which modern ćutati (chiefly Serbian) and šutjeti (chiefly Croatian) are derived. Ćutjeti, in itself, also means "to sense", although it is now poetic/archaic. He also briefly speculates how it come to the shift of the meaning (first there was an extension of the meaning, and now it is practically the full shift):

In some Bulgarian dialects we have кютя (kyutya) which sounds related to the Serbian one. I don't know where it comes from, but almost all our words starting in "кю" are from Turkish origin, so maybe this one is Turkish too.

I disagree. That does not explain Croatian šutjeti and the pretty obvious relation with ćutati 'sense', with parallels in other Slavic languages. Turkish küt and kütük do not even have a similar meaning. That Serbo-Croatian semantic drift is unusual indeed, but it's not an uncommon process in Slavic world.

I disagree. That does not explain Croatian šutjeti and the pretty obvious relation with ćutati 'sense', with parallels in other Slavic languages. Turkish küt and kütük do not even have a similar meaning. That Serbo-Croatian semantic drift is unusual indeed, but it's not an uncommon process in Slavic world.

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I did not try to explain Croation šutjeti: that has been already done by iobyo:

I've only heard the perfective forms "muknuti" and "umuknuti" in Serbian - the first isn't common in literature but is heard dialectically (at least in southeastern Serbia) while the other is more common. Both are more informal than "(u)ćutati", while "ćuti!" is rude, too.

The "ol" cluster in verbs such as in "молчать" is always realised as "u" in Serbian. Another example of it is "vuk" (wolf)